Diagonal Lemma

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nomadreid
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Main Question or Discussion Point

The diagonal lemma (which can be used to prove the Gödel-Tarski Theorem, among others) apparently goes, in a nutshell,

suppose you have a one-place formula A(.) with domain the set of codes of sentences.
I use quotation marks in this way : "M" is the code of M.
Define the 2-place relation sub:
If x ="f(.)", then sub(x,x) = "f(x)"
Define B(x) as A(sub (x,x))
Define S as B("B(x)")
retracing, S is true iff A("S") is.

Nice. What I do not understand is that the proof notes that S and A("S") are equivalent but not the same. It appears to me from the construction that they are the same. What am I missing?
Thanks.
 

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